New Journal of Physics
The open–access journal for physics
Inclined to splash: triggering and inhibiting
a splash with tangential velocity
James C Bird� Scott S H Tsai and Howard A Stone1
School of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA
E-mail: [email protected]
New �ournal of Physics 11 (2009) 063017 (10pp)
Received 5 March 2009
Published 12 June 2009
Online at http://www.njp.org/
doi:10.1088/1367-2630/11/6/063017
Abstract. When a liquid drop impacts a smooth, solid, dry surface, the drop
forms a radially spreading lamella, which can lead to a splash. Previous studies
have focused almost exclusively on impacts perpendicular to a surface; yet it
is common for drops to impact on angled or moving surfaces. The asymmetry
of such impacts leads to an azimuthal variation of the ejected rim, and under
certain conditions only part of the rim breaks up to form droplets. We show that
the tangential component of impact can act to enhance or suppress a splash. We
develop a new model to predict when this type of splashing will occur. The model
accounts for our observations of the effects of tangential velocity and agrees well
with previous experimental data.
Contents
1� Introduction
2� Experiments for the splashing threshold
3� A model for splashing with tangential boundary motion
4� Comparison of experiments and our model
5� Discussion and conclusion
Acknowledgments
References
1
2
2
4
7
7
9
9
Author to whom any correspondence should be addressed.
New �ournal of Physics 11 (2009) 063017
1367-2630/09/063017+10�30.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
1. Introduction
Upon impact, liquid drops can rapidly spread, splash, or bounce. Splashing remains the least
understood of these three responses, in part because there are numerous ways in which drops
can break apart [1]–[3]. Here we focus on the impact of drops on smooth surfaces that are rigid
and dry. For these conditions, splashing occurs when the rim of the spreading drop becomes
airborne, often forming a corona (figure 1). Capillarity subsequently disintegrates the airborne
film into satellite droplets, which are crucial to atomization, but are often detrimental in coating
processes, such as ink jet printing and pesticide delivery [2]. Most studies of splashing have
focused on the impact of drops perpendicular to stationary surfaces [1, 2]. Nevertheless, in many
applications, the target surface is either moving relative to the drop, such as raster printing,
or inclined, such as leaves on a tree [4]. In this paper, we investigate experimentally and via
modeling, how tangential velocity affects splashing on a dry, smooth surface.
The transition from spreading (figure 1(a)) to splashing (figure 1(d)) is often referred to as
the splashing threshold. Below the splashing threshold, a radial jet, or lamella, rapidly spreads
along the dry surface. Above the threshold, the lamella lifts off of the surface, often within the
first observable instances, and breaks into satellite droplets. Even though images of splashing on
dry surfaces appear similar to images of splashing on pre-wet surfaces, the dynamics between
these two conditions are fundamentally distinct, in part due to the existence of a contact
line where the solid, liquid and surrounding gas meet. Therefore, proposed mechanisms for
splashing on pre-wet surfaces, such as the propagation of kinematic discontinuities [5], are
rarely applicable to initially dry surfaces without additional assumptions.
The splashing threshold on dry surfaces has been shown to depend on the surface
roughness [6, 7], the properties of the surrounding gas [8], and the compliance of the solid
substrate [9]. Past experiments have found that the splashing threshold follows the empirical
relation W e Re1/2 � K , where K is a constant that depends on external parameters such as
2
surface roughness [6, 10]. Here, the Weber and Reynolds numbers are defined as W e � ρVγ R
and Re � VνR , respectively, where V is the impact velocity of the drop, R is the drop radius, ρ is
the liquid density, γ is the surface tension and ν is the kinematic viscosity. While this threshold
has been rationalized using theoretical arguments [2, 11, 12], few of these theoretical arguments
easily generalize to account for tangential velocity.
Although numerous studies have documented the effects of tangential velocity on drop
impact [13]–[18], only a few have reported the effects on splashing on dry surfaces. These
experiments have concluded that on an incline, the splashing threshold follows the familiar form,
W e Re1/2 � K , provided that the impact velocity is replaced with the normal velocity [6, 10].
Significant tangential velocity can also lead to asymmetric splashing, a phenomenon in which
one side of the drop spreads and the other splashes [19]. Here, we quantify the asymmetric
splashing behavior and, by treating the sides of the drop separately, develop a physical model
that not only agrees well with our experimental data, but also provides insight into the early
time dynamics of splashing.
2. Experiments for the splashing threshold
We experimentally measure the impact dynamics of millimeter-sized ethanol drops (ρ �
790 kg m−3 , ν � 1.2 × 10−6 m2 s−1 and γ � 23 mN m−1 ) on to a dry, smooth, aluminum surface.
Drops are released from a syringe located between 3 and 67 cm above the surface. The drop
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
3
Vn = 2.7 m s−1
Vn = 1.6 m s−1
(a)
(d)
1 mm
Vt = 0 m s−1
Vt = 0 m s−1
(e)
(b)
–2.4 m s−1
2.4 m s−1
(f )
(c)
–5.4 m s−1
8.2 m s−1
Figure 1. A component of velocity tangent to the surface can both induce and
suppress splashing. Arrows indicate the direction of motion of the substrate.
(a) No splash occurs when a millimeter-sized ethanol drop impacts normally
at 1.2 m s−1 upon a flat and stationary aluminum disk. (b) As the substrate moves
tangentially (here 2.4 m s−1 ), the portion of the lamella moving opposite the
substrate begins to splash. (c) At even higher tangential velocity (5.4 m s−1 ) an
asymmetric corona splash develops. (d) Above the splashing threshold, increased
tangential velocity can act to reduce (e) and eventually suppress (f) a portion of
the splash. Images (a)–(c) and (d)–(f) were captured, respectively, 1.3 and 0.5 ms
after impact.
size, impact velocity, and subsequent dynamics are measured using a Phantom V7 high-speed
camera recording at 15 000 frames per second. We generate a tangential velocity Vt by either
moving or inclining the surface relative to the drop. To move the surface at a uniform velocity,
we attach an aluminum disk to an analog-controlled electric motor and rotate it so that the
tangential velocity 6 centimeters off-center ranges from 0 to 21 m s−1 . For the experiments
with a stationary inclined plane, the surface is inclined at angles between 0 and 50◦ from the
horizontal.
We find that below the splashing threshold (figure 1(a)), the ethanol drop spreads across the
aluminum surface. When the tangential velocity is increased, the portion of the lamella moving
in the same direction as the substrate continues to spread, whereas the portion of the lamella
moving opposite to the substrate delaminates and splashes (figures 1(b) and (c)). We refer to
this symmetry breaking, when one side of the drop spreads while the other side splashes, as
asymmetric splashing. We find similar dynamics above the splashing threshold (figure 1(d)).
Increased tangential velocity acts to suppress the splash from the lamella moving in the same
direction as the applied tangential velocity (figures 1(e) and (f)).
The impact dynamics were similar for both the inclined and moving surfaces for a
given normal and tangential velocity. A phase plot of the splashing threshold quantitatively
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
4
4�0
3�5
Vn (m s–1)
3�0
2�5
2�0
Spreading
Splashing
1�5
1�0
0�5
0
–25 –20 –15 –10
–5
0
5
10
15
20
25
–1
Vt (m s )
Figure 2. For a given normal drop velocity Vn , increasing the tangential velocity
towards the lamella (negative Vt ) instigates splashing, causing a spreading
lamella (◦, �) to lift off the surface (•, �). Similarly, increasing the tangential
velocity away from the lamella (positive Vt ) inhibits splashing. The symbols
denote whether the surface is moving (◦, •) or inclined (�, �). Here the radius
of the drops are R � 0.88 mm and the dashed line is a guide for the eye.
Representative error bars indicate the size of the 95� confidence intervals at two
locations; as the tangential velocity strongly depends on where the drop impacts
the disk, the uncertainty increases with both disk speed and drop height.
demonstrates the effects of the tangential velocity (figure 2). Our results agree with previous
observations that there is a critical velocity separating spreading and splashing along Vt � 0.
Additionally, the data quantify how the splashing threshold changes with tangential velocity.
When each drop impacts the substrate, we record both the dynamics of the lamella toward
(Vt > 0) and away (Vt < 0) from the direction of the substrate motion. This characterization is
responsible for the symmetry in the position of the points in figure 2.
Special attention should be given to the range of tangential velocities needed to
significantly influence the splashing threshold. For example, shock waves in the liquid have been
shown to be important in lamellar formation [20], though at much higher impact speeds than
utilized here. More recently, experiments and theory have linked splashing to the compressibility
of the surrounding gas at early times [8, 21]. Nevertheless, the results in figure 2 demonstrate
that tangential velocities well below the speed of sound in both liquid and gas are able to have
a significant influence on the splashing behavior: the lamella speed traveling tens of meters
per second, rather than hundreds of meters per second, affects the splashing threshold. The
implication is that the mechanisms responsible for splashing in our study occur when the lamella
is traveling on the order of tens of meters per second.
3. A model for splashing with tangential boundary motion
While there exist several models for splashing, these models do not easily generalize to account
for tangential velocity. Additionally, the existing models that have been proposed for the
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
5
(a)
(b)
V�
R
Vn
δ �
a
��
L
h
h
Vt
L
r
Figure 3. (a) At early times, we assume that the portion of drop (dark
gray) that has not contacted the rigid solid is unperturbed and translates at
constant velocity Vn . The portion of the drop that has contacted the rigid solid
(region ∗ ) is redistributed into a lamella around the drop with length L and
thickness h (region ∗∗ ). (b) We model the lamella moving outward on a moving
surface. In our model, we assume that the important parameters are the bulk
lamella velocity V� , the velocity of the surface and surrounding boundary layer
Vt , the lamella thickness h and length L.
early time lamella dynamics, are either not appropriate for dry surfaces [11] or atmospheric
pressures [8]. Therefore, instead of generalizing an existing approach, we now introduce a
model of the lamella dynamics that builds on ideas from previous studies. The model is based
upon our observations that the mechanism behind splashing depends on the dynamics of the
lamella relative to the target surface or the surrounding air rather than the dynamics relative
to the drop. We describe the lamella with a characteristic length along the flow direction L,
thickness h and velocity V� . In addition, we model the surrounding substrate and gas as moving
with velocity Vt (figure 3). All parameters have positive values except for Vt , which is positive
when the substrate moves in the same direction as the lamella and negative when the substrate
moves in the opposite direction. The air boundary layer over the spinning plate is estimated
using the kinematic viscosity of air ν � 1.5 × 10−5 m2 s−1 and the maximum
angular velocity
√
of the plate ω � 5000 rpm. As the thickness of this boundary layer ν/ω > 400 µm is much
larger than the initial lamella thickness h < 50 µm, we assume that the surrounding solid and
gas are moving at the same speed.
Similar to previous models [2, 5, 11, 12], we believe that any instabilities in the film
will be described by inertial dynamics. As the inertia of the film relative to the substrate
increases, the film may destabilize, overshoot small changes in the surface topography, and
become airborne [22]. At the same time, these instabilities are regulated by capillarity. In other
words, film deformations caused by substrate or gas-related perturbations increase the interfacial
surface area. Only when the kinetic energy of the film relative to the surroundings is much
greater than the surface energy preventing deformations will the film become unstable and lead
to a splash. In our model, the kinetic energy of the lamella relative to the surroundings scales
as ρ�V� − Vt )2 L 2 h (see figure 3). Similarly, the increase in surface energy due to deformation
scales as γ h L. For splashing to occur, the kinetic energy must be much greater than the change
in surface energy, or
ρ�V� − Vt )2 L
> C � 1,
γ
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
(1)
6
where the value of C depends on the properties of the surroundings. The focus of our model
is on the liquid; since we do not consider variations to the solid or surrounding gas, C is a
constant much greater than one. Nonetheless, we expect that any additional effects that increase
the deformation of the lamella, such as surface roughness or atmospheric pressure, would
decrease C.
It is generally apparent whether or not a drop will splash as soon as the lamella is visible, as
small perturbations are observable in a destabilized lamella. For a 1 mm radius drop of ethanol
with an impact velocity of 2 m s−1 , the lamella first appears at about 10 µs [9]. The lamella
interactions occur so rapidly after contact that accurate measurements of the lamella dimensions
and speed are currently limited by the spatial and temporal resolution of high-speed cameras.
Instead, we estimate V� , h and L from the initial conditions of drop impact.
It is typical to model the lamella size and velocity based on a redistribution of fluid due
to geometric constraints (figure 3(a)) [8, 11, 23]. In this approach all of the fluid in the drop is
assumed to be moving at the speed Vn toward the surface. Once fluid particles contact the rigid
surface, they are redistributed outward into the lamella. The consequence is that the volume
of the perturbed fluid (region ∗ in figure 3(a)) is equal to the volume of the lamella (region ∗∗
in figure 3(a)). Equating the volume of the spherical cap to the volume of the annular disk in
figure 3, we find that π6 δ�3a 2 + δ 2 ) � π�r 2 − a 2 )h.
√The rate at which fluid is displaced depends on the drop speed such that δ � Vn t and
a ≈ 2Vn Rt near the point of contact, where t is the time since impact. We assume that at
very early times the lamella thickness √
scales proportional to the momentum boundary layer
thickness in the liquid, such that h � c1 νt where c1 is a scaling constant of ord(1) (e.g. [5, 8,
11]); recent experiments by de Ruiter et al (manuscript in review) suggest that the dynamics,
while approximately represented by this formula, are more complex. Using these results, we
can express the radial extent of the lamella as
�
V 2 Rt 2
r � 2Vn Rt + n√ + O��Vn t)2 ).
(2)
c1 νt
�
√
We see that in the limit that t → 0, r ≈ Vn Rt and V� � ṙ ≈ RVt n , where a dot denotes
the time derivative. The difficulty with the limit t → 0 is that the parameters in which we are
interested are no longer finite. One approach to this problem is to regularize the singularity and
produce a splashing model that is independent of time (e.g. [8]). Another approach is to note
that many of the physical assumptions break down at early times, leading to the significance of
such effects as compressibility of the liquid [20, 24] and the role of the intervening gas film [21].
Here, we assume instead that the destabilization process requires that the lamella evolves to
be distinctly separate from the drop. Provided that the contact point of the drop a�t) moves faster
than the lamella separates from the drop L�t) (figure 3(a)), any destabilization will be engulfed
in the drop. It is only when L grows at the same rate or faster than a that we have a distinct
lamella. Recalling that r � L�t) + a�t) and a 2 � 2Vn Rt, we can see from (2) that L�tc ) ≈ a�tc )
at the time
ν
tc ≈ 2 .
(3)
Vn
Equation (3), based on the assumed viscous dependence in (2), predicts that for an ethanol
drop falling at 1 m s−1 , there would be a distinct lamella on the order of 1 µs after contact. This
estimate is consistent with previous measurements [9]. Since both L and V� depend on time,
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
7
the splashing criterion (1) should depend on time as well. The left-hand side of (1) decreases
with time, therefore our model predicts that splashing occurs at the earliest time physically
appropriate, which is on the order of tc . This consequence of the model is also consistent with
experiments, which show that splashing develops when the lamella is first observable [8, 9]. At
tc our model predicts the lamellar length and velocity to be
�
L�tc ) ∼ Vn Rtc � c2 R Re−1/2 ,
(4a)
V� �tc ) ∼ L̇ � c3 Vn Re1/2 ,
(4b)
where the Reynolds number Re � VnνR and c2 and c3 are scaling constants of order one.
Substituting the scaling relations for L and V� into (1), the splashing threshold expression
can be re-expressed as
�
�
Vt −1/2 2
1/2
>K,
(5)
1 − k Re
W e Re
Vn
2
where the Weber number W e � ρVγn R , K � c2Cc3 2 and k � 1/c3 . We expect k to be order one and
K to be much greater than one; K has been previously reported to be between 1180 and 19 200
for various, relatively smooth surfaces [6, 10].
4. Comparison of experiments and our model
In the case that the substrate is stationary and flat, VVnt � 0, we predict that splashing will occur
when W e Re1/2 > K , a result that is consistent with previous splashing data [6, 10]. In these past
experiments K has been shown to decrease with surface roughness. We would expect a similar
qualitative behavior in our model as well, although the functional form is beyond the scope of
this paper.
Our approach makes testable predictions for the splashing behavior when the surface is
either inclined or moving relative to the drop trajectory. In figure 4, we plot our experimental
results for three sizes of drops following the scaling suggested by equation (5). Specifically, we
nondimensionalize the horizontal axis as �Vt /Vn )Re−1/2 , and the vertical axis as W e Re1/2 /K .
With this scaling, the data points, which correspond to whether or not a drop splashed, collapse
into distinct regions independent of the drop size. The model prediction agrees well with the
experimental data with k � 2.5 and K � 5700 (solid line in figure 4), values that are consistent
with the model (k is ord(1) and K � 1) and previous experiments [6, 10].
5. Discussion and conclusion
When one side of the drop encounters a positive tangential velocity, the other side of the drop
encounters an equal but negative tangential velocity. Therefore, it is natural to inquire how the
drop behaves globally in the presence of tangential velocity from any direction. Based on the
magnitude of the tangential velocity, there are three behaviors: the lamella will spread in all
directions, splash in all directions, or asymmetrically splash (figure 5). As expected, when there
is no tangential velocity, the drops either spread or splash in all directions. However, when the
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
8
4�5
4�0
�
We Re��2 1 − k Vt �Vn Re���2
�2
=K
We Re��2�K
3�5
3�0
Spreading
Splashing
2�5
2�0
1�5
1�0
0�5
–0�4
–0�2
0
0�2
Vt�Vn Re−��2
0�4
0�6
Figure 4. Experimental results for ethanol drops with different radii, R �
0.88 mm (◦, •), 1.3 mm (�, �) and 1.7 mm (�, �), approximately collapse on
to a master curve when rescaled according to our theory. The model prediction
(solid line) agrees well with the experimental data when k � 2.5 and K � 5700
in equation (5). Representative error bars are placed at three locations.
4�0
Splashing 3�5
both sides
3�0
We Re��2�K
Asymmetric splashing
2�5
2�0
1�5
1�0
Spreading
both sides 0�5
0
0
0�1
0�2
0�3
0�4
0�5
�Vt� �Vn Re−��2
Figure 5. Experimental results from figure 4 recast to show global drop behavior.
The model predicts three behaviors based on the magnitude of the tangential
velocity: the lamella will spread in all directions (white symbols), splash in
all directions (black symbols) or asymmetrically splash (gray symbols). Here
K � 5700.
New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)
9
symmetry is broken through the tangential velocity, the transition bifurcates to create a new
behavior: asymmetric splashing.
In this paper, we have documented the effect of tangential velocity on drop splashing.
The experiments demonstrate that sufficient tangential velocity can either trigger or inhibit
splashing on a portion of the drop. We develop a physical mechanism for splashing based on the
rate of growth of the lamella. Predictions from the model agree well with the experimental
observations. These findings are not only relevant to drop impact on moving and inclined
surfaces, but also provide insight into the fundamentals of splashing.
Acknowledgments
We thank M Brenner, L Courbin and R Pepper for helpful discussions. We also thank the NSF
via the Harvard MRSEC (DMR-0820484) for support of this research.
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New �ournal of Physics 11 (2009) 063017 (http://www.njp.org/)